OK, this came about as a result of the conversation in SCALES COURSE in this same sub-forum when White Noise asked about how to pick sounds.

The SCALES COURSE involves discussions about Intervals, and it's not really about harmony quite directly, but I discussed how intervals eventually leads to an understanding of harmony by knowing why distances from notes are as they are in intervals for a balance of propagation over the spectrum from Root to the Octave.

(I hope I didn't lose anyone there yet )

I'm running a bit short on time, so I'll try to be as concise as possible and let my fingers rip the keyboard.

Typically a lesson in Harmony is taught, like scales, from a baseline of degrees.
That's great and all, and degrees are VERY important concepts to understand, but I don't feel that a degree-centered approach to harmony really SMACKS you in the head obviously as to WHY the specific notes are "pleasing" or "not pleasing".

Often enough I've noticed that a reaction to degree standard harmony discussions ends up either leaving someone mostly still confused as to the WHY, but knowing HOW (at least to some level), and spins off others into a devil's advocate position of thinking that aesthetics are relative so the whole harmony business isn't worth much.

What's missing, I think, is a CONCRETE and tangible way of seeing the sound so that you can understand WHY that set of notes is or is not considered traditionally "pleasant".

Often, harmony is taught from a perspective of things like the Root and the Perfect 5th, the 3rd, 6th, the minor 3rd, the 7th, etc...
Chord building, essentially.

But when you hear someone say that the diminished triad is...well..diminished, you don't really understand WHY it is. You try to describe it as something along the lines of "it sounds______" insert failed attempt to describe subjective properties of aesthetics objectively.
Or, you might attempt to describe it by formula of degrees, speaking of how the notes were flattened from their natural position in some manner.

Neither really gets at the UNDERSTANDING of the cause.
The cause is the sound itself as a physical property; the WAVE of the sound.

Before, I showed a couple of wave patterns just layered over each other.
That's nice and all, but it can be challenging to see why a combination beyond two notes are harmonious or not; or even with 2 notes, it can be challenging to initially spot the harmonious wave propagation.

Now, I can take a set of waves and add them together and that will give you a somewhat clearer idea, but it still requires a set of reference charts and a really good topographical imagination to spot how it's harmonious or not.

HOWEVER, if I then take that same added set of waves and wrap it around in 360 degrees it becomes VERY, VERY, VERY easy to see when something is harmonious (in the traditional sense) or less so.

For example:
Let's just take the first site on Google that tries to explain Harmony in the typical way and use it to compare against what I'm presenting as an easier to way understand the WHY of what's going on.

A triad is, as its name implies, a chord built on thirds. The configuration of this is: root, third, and fifth—the third is a third above the root, and the fifth is a third above the third. Depending on the quality of these two third intervals, the quality of the triad changes. Below is a chart showing the different qualities of a triad you can have with the root note of C.

As you can see, there are four different types of unique triads that one can create by stacking major thirds and minor thirds. Starting from the left:

The first triad is called a diminished triad. It is so called because the fifth scale degree is lowered (or diminished) relative to a minor triad, and is made up of two minor thirds stacked upon one another.

The next triad is called the minor triad—it is created by stacking a minor third plus a major third.

After that, we have the major triad, which is created by stacking a major third plus a minor third.

Finally we have the augmented triad, the least common of the triads, which is created by stacking two major thirds.

-INSERTED COMMENT FROM THESTUMPS- he then goes into a description of degrees in basic form: 1 2 3 4 5 6 7 and gives the definition for each

The first scale degree is called the tonic—it is the home base, so to speak, and has a strong feeling of resolution; the vast majority of literature in almost every conceivable musical genre ends on the tonic.

The second scale degree is called the supertonic and has a strong tendency to resolve down a step to the tonic.

The third scale degree is called the mediant and has a relatively weak sense of resolution due to its position as the third of the C major triad.

The fourth scale degree is called the subdominant and, depending on the harmonization of the note, holds the tendency to want to resolve either down to the mediant or up a scale degree.

The fifth scale degree is called the dominant and can either have—again depending upon the harmonization—a very weak sense of resolution or a strong tendency to want to resolve to the tonic scale degree

The sixth scale degree is called the submediant and has a weak tendency to want to resolve down to the dominant.

The seventh scale degree is called the leading tone and has a very strong tendency to want to resolve up to the tonic.

Obviously, music isn’t written in only one or the other of these dimensions—harmony or melody—but rather as a combination of these two. Because of this, triads can be built upon each of the different scale degrees. Below is a chart depicting the quality of each of these resultant chords.

The roman numerals under each chord depict the quality of each chord. Upper case numerals designate major triads, lower case designate minor triads and the lowercase with the “º” designates a diminished triad.

-INSERTED COMMENT FROM THESTUMPS- he then concludes the opening discussion by noting that he will now go into the different styles in which triads are employed.

OK, now, thing is.
I doubt many reading along really feel like they understand WHY those tones are what they are.

So, let's take his first example set and just show you a bit differently than talking about degree definitions.
Let's actually LOOK at the wave.

For the first one, I'm going to show you the entire image so that it makes sense what's going on.
After that, I'm only going to show the WAVE PHASE SYMMETRY.

Alright.
We're going in the same order he did, so that's DIMINISHED TRIAD, MINOR TRIAD, MAJOR TRIAD, AUGMENTED TRIAD.

Now, the LAYERED INDIVIDUAL WAVES is each note's individual FREQUENCY just stacked over each other.
The RESULTING WAVE is the compound of those individual waves together.
And the WAVE PHASE SYMMETRY is that RESULTING WAVE in 360 degrees upon itself.

A QUICK NOTE: there's often a slight misalignment here and there between the end and beginning in the WAVE PHASE SYMMETRY view. It's always really obvious because it shows up as a curve with a sudden sharp end to itself. Just ignore it, it's just an artifact of the chart style in excel. It should normally wrap around upon itself.

Alright.
So that's neat, but what does it MEAN?
Well, to learn that, let's take a look at something a bit different.
Let's look at C, C#, and D all together.

See how symmetrical that is in the phase AROUND the intervals?
That's SUPER DUPER HARMONIOUS.

Generally speaking, the MORE symmetrical the pattern is of the RESULTING WAVE, the MORE HARMONIOUS it is...and that MAKES sense because what that means is that the WAVE is propagating in an equally dispersed manner and, being that we are beings OF nature, we tend to appreciate symmetrical things.
Especially, however, we REALLY appreciate things that are SYMMETRICAL with SLIGHT asymmetry to them...like the HUMAN FACE, or a TREE, or LEAF, or...well...ANYTHING on this planet.

That's a bit dry to us after a while.
It's neat here and there, but if all you did was just a bunch of octave pairings of notes...it would be far less entertaining to most ears than something with a few other notes involved in the chords.

SO!
Now that we have a BASIC understanding of how to READ this information, let's finish off the TRIADS that we were working on.

You'll notice that the one thing these TRIADS all have in common is that they have a general symmetry in SPACIAL occupation.
Some more than others.

BE AWARE - we're not looking for EXACT duplication like a mirror.
If that DOES happen, then that means whatever you are looking at is VERY harmonious (and likely a major chord of some kind).

What you ARE looking for is SPACIAL OCCUPATION EQUALLY.
See how the MINOR TRIAD isn't a MIRROR image, BUT, it is filling up the SPACE all around rather equally?
That's what we're looking for.

What we DON'T consider "pretty" or "good sounding" are things like that C, C#, D combination back at the beginning - those are easy to spot because you can see that the WAVE of sound UNEQUALLY fills part of SPACE differently than the rest by a LARGE amount. It's lopsided like a melted face.

And THAT is why things are Harmonious or not.
And that's how you PICK what to mix with your notes or not; by their phonic symmetry (major), and slight asymmetry (minor).

The last component is AMPLITUDE symmetry.
Take a C and Bb and you'll see that it is, indeed, symmetrical looking, but also that it POUNDS out some of the waves while others are far smaller, and does so at unequal distances (unlike the 5th - G, or the 7th - B, for example, which have more equal distances in shorter interval between amplitude differences in their wave propagation).

Keep in mind that these things are MOVING. So that unequal amplitude really can strike the ear hard and rather than a symmetry of the wave frequency propagation being part of the issue; you're looking at the wave's amplitude propagation being the issue. Creating a sort of "whop-whop-whop" effect at very rapid intervals.
Most folks find that unpleasant if left sustaining for a long time.

Both the flat 7th and 7th (A#/Bb and B, to C as your root) have this issue, and that's notably why they are considered as TRANSITION notes/chords, but not something you tend (traditionally) to pound on over and over and over.
It's not like a 4 chord punk band is going to hammer out a 7th chord repeatedly as the primary chord type for all of their songs...it doesn't quite work as good as that all powerful 5th chord (C, G, and octave C).

Thanks for including the tool, I'm looking forward to messing about with that!

A while back, out of curiosity, I set the ratio for an operator in an FM synth to the golden ratio and sent operator one and two through separate outs without any FM just to see what that interval would sound like. I played a C and operator two came out remarkably close to a G note (within a few hertz when C was 4-something or other). So it turns out part of the reason the 5th is so dang pleasing is that it's very close (probably the closest of any note) to the golden ratio multiplied by the tonic. So I kind of expected this to be all about 5ths, but it's cool (and helpful) to come at the math in a way that lets us see what other ratios are pleasing.

The 5th is approximately 1.49 higher than the root in frequency which is close-ish to 1.618~ but it fails to accomplish the golden ratio of a+b is to a as a is to b. If you subtract the root frequency from the 5th frequency, and then subtract the octave frequency from the 5th frequency, then you now have a and b distances.
Distance A+B frequencies / A will not equal A/B as a golden ratio does.

For example:
C=16.35
G=24.5
C8=32.7

G-C=8.15
C8-G=8.2
(G-C)+(C8-G)=16.35

(G-C)/(C8-G)=0.99~
((G-C)+(C8-G))/(G-C)=2.006~

So the frequency range itself in distance isn't phi ratio properly.

I've seen some try interval count to equal phi, but those attempts tend to just use the raw interval value rather than the distance. For example using 12/7 instead of 12 - 7 to get b and then say
a: 7
b: 5
a+b: 12

a/b:1.4~
(a+b)/a:1.7~

So the interval idea doesn't produce phi either.

The self-similar nature of phi is fascinating but with sound, that would produce too similar and symmetrical of sound to be interesting.

When you start looking at chords and their propagations you start to spot that nice strong chords have a slight asymmetry to them; something phi is incapable of.